No need for a grid: Adaptive fully-flexible gaussians for the time-dependent Schr\"odinger equation

• URL: http://arxiv.org/abs/2207.00271v2
• Date: Wed, 27 Jul 2022 19:51:35 GMT
• Title: No need for a grid: Adaptive fully-flexible gaussians for the time-dependent Schr\"odinger equation
• Authors: Simen Kvaal, Caroline Lasser, Thomas Bondo Pedersen, Ludwik Adamowicz
• Abstract summary: Linear combinations of complex gaussian functions are shown to be an extremely flexible representation for the solution of the Schr"odinger equation in one spatial dimension. We present a scheme based on the method of vertical lines, or Rothe's method, for propagation of such wavefunctions. This paves the way for accurate and affordable solutions of the time-dependent Schr"odinger equation for multi-atom molecules beyond the Born--Oppenheimer approximation.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Linear combinations of complex gaussian functions, where the nonlinear parameters are allowed to vary, are shown to be an extremely flexible representation for the solution of the time-dependent Schr\"odinger equation in one spatial dimension. Propagation of such wavefunctions using the Dirac--Frenkel variational principle is notoriously hard, and we present instead a scheme based on the method of vertical lines, or Rothe's method. We apply the method to a simple test system mimicking an atom subject to an extreme laser pulse, producing complicated ionization dynamics. The scheme is shown to perform very well on this model. Since the propagation method can be formulated entirely in terms of gaussian integrals and expectation values, we eliminate the need for large grids using only a handful of gaussian functions but with the same accuracy. This paves the way for accurate and affordable solutions of the time-dependent Schr\"odinger equation for multi-atom molecules beyond the Born--Oppenheimer approximation.

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